3.1941 \(\int \frac{1}{\left (a+\frac{b}{x^2}\right )^{3/2} x^8} \, dx\)

Optimal. Leaf size=95 \[ -\frac{15 a^2 \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )}{8 b^{7/2}}+\frac{15 a \sqrt{a+\frac{b}{x^2}}}{8 b^3 x}-\frac{5 \sqrt{a+\frac{b}{x^2}}}{4 b^2 x^3}+\frac{1}{b x^5 \sqrt{a+\frac{b}{x^2}}} \]

[Out]

1/(b*Sqrt[a + b/x^2]*x^5) - (5*Sqrt[a + b/x^2])/(4*b^2*x^3) + (15*a*Sqrt[a + b/x
^2])/(8*b^3*x) - (15*a^2*ArcTanh[Sqrt[b]/(Sqrt[a + b/x^2]*x)])/(8*b^(7/2))

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Rubi [A]  time = 0.15285, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{15 a^2 \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )}{8 b^{7/2}}+\frac{15 a \sqrt{a+\frac{b}{x^2}}}{8 b^3 x}-\frac{5 \sqrt{a+\frac{b}{x^2}}}{4 b^2 x^3}+\frac{1}{b x^5 \sqrt{a+\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^2)^(3/2)*x^8),x]

[Out]

1/(b*Sqrt[a + b/x^2]*x^5) - (5*Sqrt[a + b/x^2])/(4*b^2*x^3) + (15*a*Sqrt[a + b/x
^2])/(8*b^3*x) - (15*a^2*ArcTanh[Sqrt[b]/(Sqrt[a + b/x^2]*x)])/(8*b^(7/2))

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Rubi in Sympy [A]  time = 14.7964, size = 87, normalized size = 0.92 \[ - \frac{15 a^{2} \operatorname{atanh}{\left (\frac{\sqrt{b}}{x \sqrt{a + \frac{b}{x^{2}}}} \right )}}{8 b^{\frac{7}{2}}} + \frac{15 a \sqrt{a + \frac{b}{x^{2}}}}{8 b^{3} x} + \frac{1}{b x^{5} \sqrt{a + \frac{b}{x^{2}}}} - \frac{5 \sqrt{a + \frac{b}{x^{2}}}}{4 b^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**2)**(3/2)/x**8,x)

[Out]

-15*a**2*atanh(sqrt(b)/(x*sqrt(a + b/x**2)))/(8*b**(7/2)) + 15*a*sqrt(a + b/x**2
)/(8*b**3*x) + 1/(b*x**5*sqrt(a + b/x**2)) - 5*sqrt(a + b/x**2)/(4*b**2*x**3)

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Mathematica [A]  time = 0.0927381, size = 111, normalized size = 1.17 \[ \frac{\sqrt{b} \left (15 a^2 x^4+5 a b x^2-2 b^2\right )+15 a^2 x^4 \log (x) \sqrt{a x^2+b}-15 a^2 x^4 \sqrt{a x^2+b} \log \left (\sqrt{b} \sqrt{a x^2+b}+b\right )}{8 b^{7/2} x^5 \sqrt{a+\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x^2)^(3/2)*x^8),x]

[Out]

(Sqrt[b]*(-2*b^2 + 5*a*b*x^2 + 15*a^2*x^4) + 15*a^2*x^4*Sqrt[b + a*x^2]*Log[x] -
15*a^2*x^4*Sqrt[b + a*x^2]*Log[b + Sqrt[b]*Sqrt[b + a*x^2]])/(8*b^(7/2)*Sqrt[a
+ b/x^2]*x^5)

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Maple [A]  time = 0.013, size = 94, normalized size = 1. \[ -{\frac{a{x}^{2}+b}{8\,{x}^{7}} \left ( -15\,{b}^{3/2}{x}^{4}{a}^{2}+15\,\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{a{x}^{2}+b}+b}{x}} \right ) \sqrt{a{x}^{2}+b}{x}^{4}{a}^{2}b-5\,{b}^{5/2}{x}^{2}a+2\,{b}^{7/2} \right ) \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}{b}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2)^(3/2)/x^8,x)

[Out]

-1/8*(a*x^2+b)*(-15*b^(3/2)*x^4*a^2+15*ln(2*(b^(1/2)*(a*x^2+b)^(1/2)+b)/x)*(a*x^
2+b)^(1/2)*x^4*a^2*b-5*b^(5/2)*x^2*a+2*b^(7/2))/((a*x^2+b)/x^2)^(3/2)/x^7/b^(9/2
)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^(3/2)*x^8),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.258085, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (a^{3} x^{5} + a^{2} b x^{3}\right )} \sqrt{b} \log \left (\frac{2 \, b x \sqrt{\frac{a x^{2} + b}{x^{2}}} -{\left (a x^{2} + 2 \, b\right )} \sqrt{b}}{x^{2}}\right ) + 2 \,{\left (15 \, a^{2} b x^{4} + 5 \, a b^{2} x^{2} - 2 \, b^{3}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{16 \,{\left (a b^{4} x^{5} + b^{5} x^{3}\right )}}, \frac{15 \,{\left (a^{3} x^{5} + a^{2} b x^{3}\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{-b}}{x \sqrt{\frac{a x^{2} + b}{x^{2}}}}\right ) +{\left (15 \, a^{2} b x^{4} + 5 \, a b^{2} x^{2} - 2 \, b^{3}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{8 \,{\left (a b^{4} x^{5} + b^{5} x^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^(3/2)*x^8),x, algorithm="fricas")

[Out]

[1/16*(15*(a^3*x^5 + a^2*b*x^3)*sqrt(b)*log((2*b*x*sqrt((a*x^2 + b)/x^2) - (a*x^
2 + 2*b)*sqrt(b))/x^2) + 2*(15*a^2*b*x^4 + 5*a*b^2*x^2 - 2*b^3)*sqrt((a*x^2 + b)
/x^2))/(a*b^4*x^5 + b^5*x^3), 1/8*(15*(a^3*x^5 + a^2*b*x^3)*sqrt(-b)*arctan(sqrt
(-b)/(x*sqrt((a*x^2 + b)/x^2))) + (15*a^2*b*x^4 + 5*a*b^2*x^2 - 2*b^3)*sqrt((a*x
^2 + b)/x^2))/(a*b^4*x^5 + b^5*x^3)]

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Sympy [A]  time = 27.7637, size = 102, normalized size = 1.07 \[ \frac{15 a^{\frac{3}{2}}}{8 b^{3} x \sqrt{1 + \frac{b}{a x^{2}}}} + \frac{5 \sqrt{a}}{8 b^{2} x^{3} \sqrt{1 + \frac{b}{a x^{2}}}} - \frac{15 a^{2} \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} x} \right )}}{8 b^{\frac{7}{2}}} - \frac{1}{4 \sqrt{a} b x^{5} \sqrt{1 + \frac{b}{a x^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**2)**(3/2)/x**8,x)

[Out]

15*a**(3/2)/(8*b**3*x*sqrt(1 + b/(a*x**2))) + 5*sqrt(a)/(8*b**2*x**3*sqrt(1 + b/
(a*x**2))) - 15*a**2*asinh(sqrt(b)/(sqrt(a)*x))/(8*b**(7/2)) - 1/(4*sqrt(a)*b*x*
*5*sqrt(1 + b/(a*x**2)))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} x^{8}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^(3/2)*x^8),x, algorithm="giac")

[Out]

integrate(1/((a + b/x^2)^(3/2)*x^8), x)